Group with operators

In mathematics, more specifically in abstract algebra, a group with operators or Ω-group is a group with a set of group endomorphisms.

Definition

A group with operators ${\displaystyle (G,\Omega )}$  is a group ${\displaystyle G}$  together with a family of functions ${\displaystyle \Omega }$ 

${\displaystyle \omega :G\to G\quad \omega \in \Omega }$ 

which are distributive with respect to the group operation. The elements of ${\displaystyle \Omega }$  are called homotheties of ${\displaystyle G}$ .

We denote the image of a group element ${\displaystyle g}$  under a function ${\displaystyle \omega }$  with ${\displaystyle g^{\omega }}$ . The distributivity can then be expresses as

${\displaystyle (g\circ h)^{\omega }=g^{\omega }\circ h^{\omega }\quad \forall \omega \in \Omega ,\forall g,h\in G}$ .

A subgroup ${\displaystyle S}$  of ${\displaystyle G}$  is called stable subgroup,${\displaystyle \Omega }$ -subgroup or ${\displaystyle \Omega }$  invariant subgroup if it respects the hometheties, that is

${\displaystyle \forall s\in S,\forall \omega \in \Omega :s^{\omega }\in S}$ 

Notes

A group with operators is a mapping

${\displaystyle \Omega \rightarrow \operatorname {End} _{\mathbf {Grp} }(G)}$ ,

where ${\displaystyle \mathbf {Grp} }$  is the category of groups and ${\displaystyle \operatorname {End} _{\mathbf {Grp} }(G)}$  is the set of group endomorphisms of ${\displaystyle G}$ .

Examples

• For a group ${\displaystyle G}$  ${\displaystyle (G,\emptyset )}$  is trivially a group with operators
• Given a ${\displaystyle \mathbb {K} }$ -vector space ${\displaystyle V}$  then ${\displaystyle (V,\mathbb {K} )}$  is a group with operators.

References

• Bourbaki, Nicolas (1998). Elements of Mathematics : Algebra I Chapters 1-3. Springer-Verlag. ISBN 3540642439.